# 6 number

 0 10 20 30 40 50 60 70 80 90 >> Cardinal 6 six Ordinal 6th sixth Numeral system senary Factorization $\displaystyle 2 \cdot 3$ Divisors 1, 2, 3, 6 Roman numeral VI Unicode representation of Roman numeral Ⅵ, ⅵ prefixes hexa-/hex- (from Greek) sexa-/sex- (from Latin) Binary 110 Octal 6 Duodecimal 6 Hexadecimal 6 Hebrew ו (Vav)

6 (six) is the natural number following 5 and preceding 7.

The SI prefix for 10006 is exa (E), and for its reciprocal atto (a).

## In mathematics

Six is the second smallest composite number, its proper divisors being 1, 2 and 3. Since six equals the sum of these proper divisors, six is a perfect number. As a perfect number, 6 is related to the Mersenne prime 3, since 21(22 - 1) = 6. The next perfect number is 28. Six is also a unitary perfect number, a harmonic divisor number and a highly composite number. The next highly composite number is 12.

The smallest non-abelian group is the symmetric group S3 which has 3! = 6 elements.

S6, with 720 elements, is the only finite symmetric group which has an outer automorphism. This automorphism allows us to construct a number of exceptional mathematical objects such as the S(5,6,12) Steiner system, the projective plane of order 4 and the Hoffman-Singleton graph. A closely related result is the following theorem: 6 is the only natural number n for which there is a construction of n ismomorphic objects on an n-set A, invariant under all permutations of A, but not naturally in 1-1 correspondence with the elements of A. This can also be expressed category theoretically: consider the category whose objects are the n element sets and whose arrows are the bijections between the sets. This category has a non-trivial functor to itself only for n=6.

Six similar coins can be arranged around a central coin of the same radius so that each coin makes contact with the central one (and touches both its neighbors without a gap), but seven cannot be so arranged. This makes 6 the answer to the two-dimensional kissing number problem. The densest sphere packing of the plane is obtained by extending this pattern to the hexagonal lattice in which each circle touches just six others.

In binary code, six is 110; in ternary code six is 20; in quaternary numeral system code six is 12; in quinary six is 11; in senary six is 10; in septenary code and all codes above (such as octal, decimal and hexadecimal) six is 6. Since it is divisible by the sum of its digits in all these bases, 6 is one of the four all-Harshad numbers.

A six-sided polygon is a hexagon. Figurate numbers representing hexagons (including six) are called hexagonal numbers. Six is also an octahedral number. It is a triangular number and so is its square (36).

In base 10, 6 is a 1-automorphic number.

## The Arabic glyph

The evolution of our modern glyph for 6 appears rather simple when compared with that for the other numerals. Our modern 6 can be traced back to the Brahmin Indians, who wrote it in one stroke like a cursive lowercase e rotated 45 degrees clockwise. Gradually, the upper part of the stroke (above the central squiggle) became more curved, while the lower part of the stroke (below the central squiggle) became straighter. The Ghubar Arabs dropped the part of the stroke below the squiggle. From there, the European evolution to our modern 6 was very straightforward, aside from a flirtation with a glyph that looked more like an uppercase G.

On the seven-segment displays of calculators and watches, 6 is usually written with six segments. Some historical calculator models use just five segments for the 6, by omitting the top horizontal bar. This glyph variant has not caught on. For calculators that can display results in hexadecimal, a 6 that looks like a B is not practical.

In fonts with text figures, 6 usually has an ascender, for example, File:TextFigs036.png.

## In science

In astronomy,

Messier object M6, a magnitude 4.5 open cluster in the constellation Scorpius, also known as the Butterfly Cluster.
The New General Catalogue object NGC 6, a spiral galaxy in the constellation Andromeda
The Saros number of the solar eclipse series which began on -2691 March 16 and ended on -1393 May 3. The duration of Saros series 6 was 1298.1 years, and it contained 73 solar eclipses.
The Saros number of the lunar eclipse series which began on -2642 July 25 and ended on -1091 February 10. The duration of Saros series 6 was 1550.6 years, and it contained 87 lunar eclipses.

## In music

• The number of strings on a standard guitar.
• The number of basic holes or keys on most woodwind instruments (e.g., pennywhistle, clarinet, saxophone, bassoon). These holes or keys are usually not given numbers or letters in the fingering charts.
• "Six Geese a Laying" were given as a present on the sixth day in the popular Christmas carol The Twelve Days of Christmas.

## In technology

• On most phones, the 6 key is associated with the letters M, N, and O, but on the BlackBerry it is the key for J and K.

## In other fields

Six is:

Hexa is Greek for "six". Thus:

The prefix "hexa-" also occurs in the systematic name of many chemical compounds, such as "hexamethyl".

Sex- is a Latin prefix meaning "six". Thus:

• A group of six musicians is called a sextet.
• Six babies delivered in one birth are sextuplets. The first set of sextuplets of whom all six survived are the Dilley sextuplets.
• People with sexdactyly have six fingers on each hand.
• The measuring instrument called a sextant got its name because its shape forms one sixth of a whole circle.